On generalization of Breuil--Schraen's $\mathscr{L}$-invariants to $\mathrm{GL}_n$

Let $p$ be prime number and $K$ be a $p$-adic field. We systematically compute the higher $\mathrm{Ext}$-groups between locally analytic generalized Steinberg representations (LAGS for short) of $\mathrm{GL}_n(K)$ via a new combinatorial treatment of some spectral sequences arising from the so-called Tits complex. Such spectral sequences degenerate at the second page and each $\mathrm{Ext}$-group admits a canonical filtration whose graded pieces are terms in the second page of the corresponding spectral sequence. For each pair of LAGS, we are particularly interested their $\mathrm{Ext}$-groups in the bottom two non-vanishing degrees. We write down an explicit basis for each graded piece (under the canonical filtration) of such an $\mathrm{Ext}$-group, and then describe the cup product maps between such $\mathrm{Ext}$-groups using these bases. As an application, we generalize Breuil's $\mathscr{L}$-invariants for $\mathrm{GL}_2(\mathbb{Q}_p)$ and Schraen's higher $\mathscr{L}$-invariants for $\mathrm{GL}_3(\mathbb{Q}_p)$ to $\mathrm{GL}_n(K)$. Along the way, we also establish a generalization of Bernstein--Zelevinsky geometric lemma to admissible locally analytic representations constructed by Orlik--Strauch, generalizing a result in Schraen's thesis for $\mathrm{GL}_3(\mathbb{Q}_p)$.